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Difference between revisions of "2013 USAJMO Problems/Problem 1"

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re there integers and such that and are both perfect cubes of integers?
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==Problem==
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Are there integers <math> a </math> and <math> b </math> such that <math> a^5b+3 </math> and <math> ab^5+3 </math> are both perfect cubes of integers?
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==Solution==
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No, such integers do not exist.  This shall be proven by contradiction, by showing that if <math>a^5b+3</math> is a perfect cube then <math>ab^5+3</math> cannot be.
 +
 
 +
Remark that perfect cubes are always congruent to <math>0</math>, <math>1</math>, or <math>-1</math> modulo <math>9</math>.  Therefore, if <math>a^5b+3\equiv 0,1,\text{ or} -1\pmod{9}</math>, then <math>a^5b\equiv 5,6,\text{ or }7\pmod{9}</math>.
 +
 
 +
If <math>a^5b\equiv 6\pmod 9</math>, then note that <math>3|b</math>.  (This is because if <math>3|a</math> then <math>a^5b\equiv 0\pmod 9</math>.)  Therefore <math>ab^5\equiv 0\pmod 9</math> and <math>ab^5+3\equiv 3\pmod 9</math>, contradiction.
 +
 
 +
Otherwise, either <math>a^5b\equiv 5\pmod 9</math> or <math>a^5b\equiv 7\pmod 9</math>.  Note that since <math>a^6b^6</math> is a perfect sixth power, and since neither <math>a</math> nor <math>b</math> contains a factor of <math>3</math>, <math>a^6b^6\equiv 1\pmod 9</math>.  If <math>a^5b\equiv 5\pmod 9</math>, then <cmath>a^6b^6\equiv (a^5b)(ab^5)\equiv 5ab^5\equiv 1\pmod 9\implies ab^5\equiv 2\pmod 9.</cmath>  Similarly, if <math>a^5b\equiv 7\pmod 9</math>, then <cmath>a^6b^6\equiv (a^5b)(ab^5)\equiv 7ab^5\equiv 1\pmod 9\implies ab^5\equiv 4\pmod 9.</cmath>  Therefore <math>ab^5+3\equiv 5,7\pmod 9</math>, contradiction.
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Therefore no such integers exist.
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 +
Amkan2022
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==Solution 2==
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We shall prove that such integers do not exist via contradiction.
 +
Suppose that <math>a^5b + 3 = x^3</math> and <math>ab^5 + 3 = y^3</math> for integers x and y. Rearranging terms gives <math>a^5b = x^3 - 3</math> and <math>ab^5 = y^3 - 3</math>. Solving for a and b (by first multiplying the equations together and taking the sixth root) gives a = <math>(x^3 - 3)^\frac{5}{24} (y^3 - 3)^\frac{-1}{24}</math> and b = <math>(y^3 - 3)^\frac{5}{24} (x^3 - 3)^\frac{-1}{24}</math>. Consider a prime p in the prime factorization of <math>x^3 - 3</math> and <math>y^3 - 3</math>. If it has power <math>r_1</math> in <math>x^3 - 3</math> and power <math>r_2</math> in <math>y^3 - 3</math>, then <math>5r_1</math> - <math>r_2</math> is a multiple of 24 and <math>5r_2</math> - <math>r_1</math> also is a multiple of 24.
 +
 
 +
Adding and subtracting the divisions gives that <math>r_1</math> - <math>r_2</math> divides 12. (actually, <math>r_1 - r_2</math> is a multiple of 4, as you can verify if <math>\{^{5r_1 - r_2 = 24}_{5r_2 - r_1 = 48}</math>. So the rest of the proof is invalid.) Because <math>5r_1</math> - <math>r_2</math> also divides 12, <math>4r_1</math> divides 12 and thus <math>r_1</math> divides 3. Repeating this trick for all primes in <math>x^3 - 3</math>, we see that <math>x^3 - 3</math> is a perfect cube, say <math>q^3</math>. Then <math>x^3 - q^3 = 3,</math> and <math>(x-q)(x^2 + xq + q^2) = 3</math>, so that <math>x - q = 1</math> and <math>x^2 + xq + q^2 = 3</math>. Clearly, this system of equations has no integer solutions for <math>x</math> or <math>q</math>, a contradiction, hence completing the proof.
 +
 
 +
Therefore no such integers exist.
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==Solution 3==
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Let <math>a^5b+3=x^3</math> and <math>ab^5+3=y^3</math>. Then, <math>a^5b=x^3-3</math>, <math>ab^5=y^3-3</math>, and <cmath>(ab)^6=(x^3-3)(y^3-3)</cmath>
 +
Now take <math>\text{mod }9</math> (recall that perfect cubes <math>\equiv -1,0,1\pmod{9}</math> and perfect sixth powers <math>\equiv 0,1\pmod{9}</math>) on both sides. There are <math>3\times 3=9</math> cases to consider on what values <math>\text{mod }9</math> that <math>x^3</math> and <math>y^3</math> take. Checking these <math>9</math> cases, we see that only <math>x^3\equiv y^3\equiv 0\pmod{9}</math> or <math>x\equiv y\equiv 0\pmod{3}</math> yield a valid residue <math>\text{mod }9</math> (specifically, <math>(x^3-3)(y^3-3)\equiv 0\pmod{9}</math>). But this means that <math>3\mid ab</math>, so <math>729\mid (ab)^6</math> so <cmath>729\mid (x^3-3)(y^3-3)\iff 729\mid (27x'^3-3)(27y'^3-3)\iff 81\mid (9x'^3-1)(9y'^3-1)</cmath> contradiction.
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==Solution 4==
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If <math>a^5b+3</math> is a perfect cube, then <math>a^5b</math> can be one of <math>5,6,7 \pmod 9</math>, so <math>(a^5b)^5 = a^{25} b^5</math> can be one of <math>5^5 \equiv 2</math>, <math>6^5 \equiv 0</math>, or <math>7^5 \equiv 4 \pmod 9</math>. If <math>a</math> were divisible by <math>3</math>, we'd have <math>a^5 b \equiv 0 \pmod 9</math>, which we've ruled out. So <math>\gcd(a,9) = 1</math>, which means <math>a^6 \equiv 1 \pmod 9</math>, and therefore <math>a^{25} b^5 \equiv ab^5 \pmod 9</math>.
 +
 
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We've shown that <math>a b^5</math> can be one of <math>0, 2, 4 \pmod 9</math>, so <math>ab^5 + 3</math> can be one of <math>3, 5, 7 \pmod 9</math>. None of these are possibilities for a perfect cube, so if <math>a^5b+3</math> is a perfect cube, <math>ab^5+3</math> cannot be.
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==Solution 5==
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As in previous solutions, notice <math>ab^5,a^5b \equiv 5,6,7 \pmod 9</math>. Now multiplying gives <math>a^6b^6</math>, which is only <math>0,1 \pmod 9</math>, so after testing all cases we find that <math>ab^5\equiv a^5b \equiv 6 \mod 9</math>. Then since <math>\phi (9) = 6</math>, <math>ab^5\equiv \frac{a}{b}\pmod 9</math> and <math>a^5b \equiv \frac{b}{a}\pmod 9</math> (Note that <math>a,b</math> cannot be <math>0\pmod 9</math>). Thus we find that the inverse of <math>6</math> is itself under modulo <math>9</math>, a contradiction.
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==Solution 6==
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I claim there are no such a or b such that both expressions are cubes.
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Assume to the contrary <math>a^5b +3</math> and <math>ab^5 + 3</math> are cubes.
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'''Lemma 1''': If <math>a^5b +3</math> and <math>ab^5 + 3</math> are cubes, then <math>ab^5, a^5b \equiv 5,7 \pmod 9</math>
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'''Proof''' Since cubes are congruent to any of <math>0, 1, -1 \pmod 9</math>, <math>ab^5,a^5b \equiv 5,6,7 \pmod 9</math>. But if <math>ab^5 \equiv 6 \pmod 9</math>, <math>3|a</math>, so <math>a^5b \equiv 0 \pmod 9</math>, contradiction. A similar argument can be made for <math>ab^5 \neq 6 \pmod 9</math>.
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'''Lemma 2''': If k is a perfect 6th power, then <math>k \equiv 0,1 \pmod 9</math>
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'''Proof''': Since cubes are congruent to <math>0, 1,  -1 \pmod 9</math>, we can square, and get 6th powers are congruent to <math>0, 1 \pmod 9</math>.
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Since <math>ab^5 \cdot a^5b = a^6 b^6 = (ab)^6</math>, which is a perfect 6th power, by lemma 2, <math>ab^5 \cdot a^5b \equiv 0,1 \pmod 9</math>.
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But, by lemma 1, <math>ab^5 \cdot a^5b \equiv 5 \cdot 5, 5 \cdot 7, 7 \cdot 7 \equiv 7, 8, 4 \pmod 9</math>.
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So, <math>ab^5 \cdot a^5b</math>, which is an integer, can't go into any of the possible residue classes modulo 9, without breaking one of these lemmas. This is a contradiction, and the proof is complete. <math>\blacksquare</math>
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-AlexLikeMath
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==Solution 7==
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Note that <math>x^3 \equiv 0,1,-1 \pmod 9</math>. So <math>a^5b+3, ab^5+3\equiv 0,1,-1\pmod{9}</math><math>\Rightarrow a^5b, ab^5 \equiv 5,6,7 \pmod 9</math>.
 +
            <math>a^5b \equiv 5 \pmod 9 \;\;\;\;\;\; ab^5 \equiv 5 \pmod 9</math>
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            <math>a^5b \equiv 6 \pmod 9 \;\;\;\;\;\; ab^5 \equiv 6 \pmod 9</math>
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            <math>a^5b \equiv 7 \pmod 9 \;\;\;\;\;\; ab^5 \equiv 7 \pmod 9</math>.
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Multiplying <math>a^5b</math> and <math>ab^5</math> we get <math>(ab)^6</math>. By substituting values for <math>a^5b</math> and <math>ab^5</math> we can see that <math>(ab)^6 \equiv 0,1,3,6,7,8 \pmod 9</math> but because cubes are not <math>3,6,7 \pmod 9</math>, and squares are not <math>-1\pmod 9</math> so <math>\boxed{(ab)^6 \equiv 0,1 \pmod 9}</math>.
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*Case I: <math>(ab)^6\equiv 0 \pmod 9</math>.This is only true when <math>3|ab</math>. If <math>3|a,b</math> then <math>a^5b+3 \equiv 3 \pmod 9</math>, which is not a perfect cube. The same can be seen with if <math>3|a</math> then <math>a^5b+3 \equiv 3 \pmod 9</math> and if <math>3|a</math> then <math>b^5a+3 \equiv 3 \pmod 9</math>. So 6 mod 9 case is eliminated from the above relations of congruences
 +
*Case II: <math>(ab)^6\equiv 1 \pmod 9</math> for this we'll take a look at the above information regarding <math>a^5b</math> and <math>ab^5</math>. If <math>a^5b \equiv 5 \pmod 9</math> then,
 +
          <math>(ab)^6=a^5b \cdot ab^5 \equiv 5(ab^5) \pmod 9 \Rightarrow ab^5 \equiv 2 \pmod 9 \Rightarrow ab^5+3 \equiv
 +
            5 \pmod 9</math>
 +
which is not possible. Similarly, if <math>a^5b \equiv 7 \pmod 9</math> then,
 +
          <math>(ab)^6=a^5b \cdot ab^5 \equiv 7(ab^5) \pmod 9 \Rightarrow ab^5 \equiv 4 \pmod 9 \Rightarrow ab^5+3 \equiv
 +
            7 \pmod 9</math>
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which is not possible. So, no such integers.
 +
 
 +
-Lakshya Pamecha
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==Note==
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After you get that <math>a^5 b \equiv 5, 6, \text{or} 7 \pmod{9}</math>, you can also bash by listing all possible residues for <math>a^5</math> and their corresponding values for <math>b</math>. At that point, you can easily see that all the solutions for <math>a</math> and <math>b</math> do not work in the other equation, meaning that there are no solutions that satisfy both equations.
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{{MAA Notice}}

Latest revision as of 01:45, 10 May 2024

Problem

Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?

Solution

No, such integers do not exist. This shall be proven by contradiction, by showing that if $a^5b+3$ is a perfect cube then $ab^5+3$ cannot be.

Remark that perfect cubes are always congruent to $0$, $1$, or $-1$ modulo $9$. Therefore, if $a^5b+3\equiv 0,1,\text{ or} -1\pmod{9}$, then $a^5b\equiv 5,6,\text{ or }7\pmod{9}$.

If $a^5b\equiv 6\pmod 9$, then note that $3|b$. (This is because if $3|a$ then $a^5b\equiv 0\pmod 9$.) Therefore $ab^5\equiv 0\pmod 9$ and $ab^5+3\equiv 3\pmod 9$, contradiction.

Otherwise, either $a^5b\equiv 5\pmod 9$ or $a^5b\equiv 7\pmod 9$. Note that since $a^6b^6$ is a perfect sixth power, and since neither $a$ nor $b$ contains a factor of $3$, $a^6b^6\equiv 1\pmod 9$. If $a^5b\equiv 5\pmod 9$, then \[a^6b^6\equiv (a^5b)(ab^5)\equiv 5ab^5\equiv 1\pmod 9\implies ab^5\equiv 2\pmod 9.\] Similarly, if $a^5b\equiv 7\pmod 9$, then \[a^6b^6\equiv (a^5b)(ab^5)\equiv 7ab^5\equiv 1\pmod 9\implies ab^5\equiv 4\pmod 9.\] Therefore $ab^5+3\equiv 5,7\pmod 9$, contradiction.

Therefore no such integers exist.

Amkan2022

Solution 2

We shall prove that such integers do not exist via contradiction. Suppose that $a^5b + 3 = x^3$ and $ab^5 + 3 = y^3$ for integers x and y. Rearranging terms gives $a^5b = x^3 - 3$ and $ab^5 = y^3 - 3$. Solving for a and b (by first multiplying the equations together and taking the sixth root) gives a = $(x^3 - 3)^\frac{5}{24} (y^3 - 3)^\frac{-1}{24}$ and b = $(y^3 - 3)^\frac{5}{24} (x^3 - 3)^\frac{-1}{24}$. Consider a prime p in the prime factorization of $x^3 - 3$ and $y^3 - 3$. If it has power $r_1$ in $x^3 - 3$ and power $r_2$ in $y^3 - 3$, then $5r_1$ - $r_2$ is a multiple of 24 and $5r_2$ - $r_1$ also is a multiple of 24.

Adding and subtracting the divisions gives that $r_1$ - $r_2$ divides 12. (actually, $r_1 - r_2$ is a multiple of 4, as you can verify if $\{^{5r_1 - r_2 = 24}_{5r_2 - r_1 = 48}$. So the rest of the proof is invalid.) Because $5r_1$ - $r_2$ also divides 12, $4r_1$ divides 12 and thus $r_1$ divides 3. Repeating this trick for all primes in $x^3 - 3$, we see that $x^3 - 3$ is a perfect cube, say $q^3$. Then $x^3 - q^3 = 3,$ and $(x-q)(x^2 + xq + q^2) = 3$, so that $x - q = 1$ and $x^2 + xq + q^2 = 3$. Clearly, this system of equations has no integer solutions for $x$ or $q$, a contradiction, hence completing the proof.

Therefore no such integers exist.

Solution 3

Let $a^5b+3=x^3$ and $ab^5+3=y^3$. Then, $a^5b=x^3-3$, $ab^5=y^3-3$, and \[(ab)^6=(x^3-3)(y^3-3)\] Now take $\text{mod }9$ (recall that perfect cubes $\equiv -1,0,1\pmod{9}$ and perfect sixth powers $\equiv 0,1\pmod{9}$) on both sides. There are $3\times 3=9$ cases to consider on what values $\text{mod }9$ that $x^3$ and $y^3$ take. Checking these $9$ cases, we see that only $x^3\equiv y^3\equiv 0\pmod{9}$ or $x\equiv y\equiv 0\pmod{3}$ yield a valid residue $\text{mod }9$ (specifically, $(x^3-3)(y^3-3)\equiv 0\pmod{9}$). But this means that $3\mid ab$, so $729\mid (ab)^6$ so \[729\mid (x^3-3)(y^3-3)\iff 729\mid (27x'^3-3)(27y'^3-3)\iff 81\mid (9x'^3-1)(9y'^3-1)\] contradiction.

Solution 4

If $a^5b+3$ is a perfect cube, then $a^5b$ can be one of $5,6,7 \pmod 9$, so $(a^5b)^5 = a^{25} b^5$ can be one of $5^5 \equiv 2$, $6^5 \equiv 0$, or $7^5 \equiv 4 \pmod 9$. If $a$ were divisible by $3$, we'd have $a^5 b \equiv 0 \pmod 9$, which we've ruled out. So $\gcd(a,9) = 1$, which means $a^6 \equiv 1 \pmod 9$, and therefore $a^{25} b^5 \equiv ab^5 \pmod 9$.

We've shown that $a b^5$ can be one of $0, 2, 4 \pmod 9$, so $ab^5 + 3$ can be one of $3, 5, 7 \pmod 9$. None of these are possibilities for a perfect cube, so if $a^5b+3$ is a perfect cube, $ab^5+3$ cannot be.

Solution 5

As in previous solutions, notice $ab^5,a^5b \equiv 5,6,7 \pmod 9$. Now multiplying gives $a^6b^6$, which is only $0,1 \pmod 9$, so after testing all cases we find that $ab^5\equiv a^5b \equiv 6 \mod 9$. Then since $\phi (9) = 6$, $ab^5\equiv \frac{a}{b}\pmod 9$ and $a^5b \equiv \frac{b}{a}\pmod 9$ (Note that $a,b$ cannot be $0\pmod 9$). Thus we find that the inverse of $6$ is itself under modulo $9$, a contradiction.

Solution 6

I claim there are no such a or b such that both expressions are cubes.

Assume to the contrary $a^5b +3$ and $ab^5 + 3$ are cubes.

Lemma 1: If $a^5b +3$ and $ab^5 + 3$ are cubes, then $ab^5, a^5b \equiv 5,7 \pmod 9$

Proof Since cubes are congruent to any of $0, 1, -1 \pmod 9$, $ab^5,a^5b \equiv 5,6,7 \pmod 9$. But if $ab^5 \equiv 6 \pmod 9$, $3|a$, so $a^5b \equiv 0 \pmod 9$, contradiction. A similar argument can be made for $ab^5 \neq 6 \pmod 9$.


Lemma 2: If k is a perfect 6th power, then $k \equiv 0,1 \pmod 9$

Proof: Since cubes are congruent to $0, 1, -1 \pmod 9$, we can square, and get 6th powers are congruent to $0, 1 \pmod 9$.


Since $ab^5 \cdot a^5b = a^6 b^6 = (ab)^6$, which is a perfect 6th power, by lemma 2, $ab^5 \cdot a^5b \equiv 0,1 \pmod 9$.

But, by lemma 1, $ab^5 \cdot a^5b \equiv 5 \cdot 5, 5 \cdot 7, 7 \cdot 7 \equiv 7, 8, 4 \pmod 9$.

So, $ab^5 \cdot a^5b$, which is an integer, can't go into any of the possible residue classes modulo 9, without breaking one of these lemmas. This is a contradiction, and the proof is complete. $\blacksquare$

-AlexLikeMath


Solution 7

Note that $x^3 \equiv 0,1,-1 \pmod 9$. So $a^5b+3, ab^5+3\equiv 0,1,-1\pmod{9}$$\Rightarrow a^5b, ab^5 \equiv 5,6,7 \pmod 9$. 

           $a^5b \equiv 5 \pmod 9 \;\;\;\;\;\; ab^5 \equiv 5 \pmod 9$
           $a^5b \equiv 6 \pmod 9 \;\;\;\;\;\; ab^5 \equiv 6 \pmod 9$
           $a^5b \equiv 7 \pmod 9 \;\;\;\;\;\; ab^5 \equiv 7 \pmod 9$.

Multiplying $a^5b$ and $ab^5$ we get $(ab)^6$. By substituting values for $a^5b$ and $ab^5$ we can see that $(ab)^6 \equiv 0,1,3,6,7,8 \pmod 9$ but because cubes are not $3,6,7 \pmod 9$, and squares are not $-1\pmod 9$ so $\boxed{(ab)^6 \equiv 0,1 \pmod 9}$.

  • Case I: $(ab)^6\equiv 0 \pmod 9$.This is only true when $3|ab$. If $3|a,b$ then $a^5b+3 \equiv 3 \pmod 9$, which is not a perfect cube. The same can be seen with if $3|a$ then $a^5b+3 \equiv 3 \pmod 9$ and if $3|a$ then $b^5a+3 \equiv 3 \pmod 9$. So 6 mod 9 case is eliminated from the above relations of congruences
  • Case II: $(ab)^6\equiv 1 \pmod 9$ for this we'll take a look at the above information regarding $a^5b$ and $ab^5$. If $a^5b \equiv 5 \pmod 9$ then,
          $(ab)^6=a^5b \cdot ab^5 \equiv 5(ab^5) \pmod 9 \Rightarrow ab^5 \equiv 2 \pmod 9 \Rightarrow ab^5+3 \equiv               5 \pmod 9$

which is not possible. Similarly, if $a^5b \equiv 7 \pmod 9$ then, 

          $(ab)^6=a^5b \cdot ab^5 \equiv 7(ab^5) \pmod 9 \Rightarrow ab^5 \equiv 4 \pmod 9 \Rightarrow ab^5+3 \equiv              7 \pmod 9$

which is not possible. So, no such integers.

-Lakshya Pamecha

Note

After you get that $a^5 b \equiv 5, 6, \text{or} 7 \pmod{9}$, you can also bash by listing all possible residues for $a^5$ and their corresponding values for $b$. At that point, you can easily see that all the solutions for $a$ and $b$ do not work in the other equation, meaning that there are no solutions that satisfy both equations.

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